= Bath cooling followed by insulated equilibration
{title2=$T_\infty=\frac1L\int_0^L T(x,t_b)\,dx$}
Let a homogeneous plate $0<x<L$ begin at uniform <temperature> $T_0$, cool with zero-temperature <Dirichlet boundary conditions> until time $t_b$, then switch to insulated <Neumann boundary conditions>. For <thermal diffusivity> $\kappa>0$, the <Fourier sine series> during cooling is $T=(4T_0/\pi)\sum_{n\geq1,\ n\text{ odd}}\sin(n\pi x/L)e^{-\kappa(n\pi/L)^2t}/n$. Integrating the insulated <heat equation> shows that its <integral average> is conserved. All nonconstant <Fourier cosine series> modes then decay, proving that the final uniform <temperature> is $T_\infty=(8T_0/\pi^2)\sum_{n\geq1,\ n\text{ odd}}e^{-\kappa(n\pi/L)^2t_b}/n^2$. Its limits are $T_0$ at zero bath time and zero at infinite bath time. Instantaneous changes of boundary condition can create initial-boundary corner incompatibilities without invalidating the positive-time solution.
Back to article page
